# Finding Initial Velocity from Rate, Distance, and Time

• bhoover05
In summary, the conversation discusses integral word problems and the specific problem of finding the initial velocity of a particle moving at a constant acceleration of 5 m/s^2 and traveling 60m in 4 seconds. The conversation suggests starting with an equation expressing velocity as a function of time and using integration to solve for the initial velocity.

#### bhoover05

Hi guys! We started integral word problems today. I am very confused and could use some help with the following;

A particle is moving along a straight line accelerating at a constant rate of 5 m/s^2. Find the initial velocity if the particle moves 60m in the first 4 seconds.

Now, I have been able to do problems similar to these in the past, however without having an equation to start off with, I am confused to where I even begin.
t=4s
distance=60m
accelerating= 5 m/s^2
initial velocity=?

Can you start by writing an equation that expresses the velocity as a function of t? Call the initial velocity $v_0$ for now, since you don't know its value.

Velocity is the time derivative of distance and acceleration is the time derivative of velocity. So, integrate the acceleration once with respect to time (and don't forget the beloved constant of integration) to get an expression for velocity and integrate velocity to get distance. That should give you enough to solve the problem.

## 1. What is the formula for finding initial velocity from rate, distance, and time?

The formula for finding initial velocity is v = d/t , where v is the initial velocity, d is the distance traveled, and t is the time taken.

## 2. How do I know which values to plug into the formula?

You will need to have the values for distance and time. The rate is typically given in the problem and can be used to calculate the distance or time if needed.

## 3. Can I use this formula for any type of motion?

Yes, this formula can be used for any type of motion as long as the initial velocity remains constant.

## 4. What units should I use for the values in the formula?

The units for distance should be consistent with the units for time. For example, if distance is given in meters, time should be given in seconds. The resulting velocity will be in the unit of distance over time (e.g. meters per second).

## 5. Is the calculated initial velocity the same as the average velocity?

Not necessarily. The initial velocity is the velocity at the beginning of the motion, while the average velocity is the total displacement over total time. These values may be the same if the motion is constant, but they can be different for non-constant motion.